Fixed domain asymptotics and consistent estimation for gaussian random field models in spatial statistics and computer experiments

FIXED DOMAIN ASYMPTOTICS AND
CONSISTENT ESTIMATION FOR GAUSSIAN
RANDOM FIELD MODELS IN SPATIAL
STATISTICS AND COMPUTER
EXPERIMENTS
WANG DAQING
(B.Sc. University of Science and Technology of China)
A THESIS SUBMITTED
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
DEPARTMENT OF STATISTICS AND APPLIED
PROBABILITY
NATIONAL UNIVERSITY OF SINGAPORE
2010
ii
ACKNOWLEDGEMENTS
I am so grateful that I have Professor Loh Wei Liem as my supervisor. He is
truly a great mentor not only in statistics but also in daily life. I would like to thank
him for his guidance, encouragement, time, and endless patience. Next, I would
like to thank my senior Li Mengxin for discussion on various topics in research. I
also thank all my friends who helped me to make life easier as a graduate student.
I wish to express my gratitude to the university and the department for supporting
me through NUS Graduate Research Scholarship. Finally, I will thank my family
for their love and support.
iii
CONTENTS
Acknowledgements ii
Summary v
List of Tables vii
List of Figures ix
List of Notations xi
Chapter 1 Introduction 1

1.1 Mat´ern Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Powered Exponential Class . . . . . . . . . . . . . . . . . . . . . . . 6
Chapter 2 Isotropic Covariance Function 11
CONTENTS iv
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Some Probability Inequalities . . . . . . . . . . . . . . . . . . . . . 21
2.4 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Tapered Covariance Functions . . . . . . . . . . . . . . . . . . . . . 40
2.6 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.7 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.7.1 Precision of Theorem 2.3 approximations for finite 𝑛 . . . . 68
2.7.2 Precision of Theorem 2.1 and 2.2 approximations for finite 𝑛 71
Chapter 3 Multiplicative Covariance Function 89
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
3.2 Quadratic Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
3.3 Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.3.1 Spectral Density Function . . . . . . . . . . . . . . . . . . . 108
3.3.2 Mean of the Periodogram . . . . . . . . . . . . . . . . . . . 124
3.3.3 Covariance and Variance of the Periodogram . . . . . . . . . 136
3.3.4 Consistent Estimation . . . . . . . . . . . . . . . . . . . . . 148
3.4 Multiplicative Mat´ern Class . . . . . . . . . . . . . . . . . . . . . . 151
3.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
Chapter 4 Conclusion 157
Bibliography 161
v
SUMMARY
Let 𝑋 : ℝ
𝑑
→ ℝ be a mean-zero Gaussian random field with covariance function

𝐶𝑜𝑣(𝑋(x), 𝑋(y)) = 𝜎
2
𝐾
𝜃
(x −y), ∀ x, y ∈ ℝ
𝑑
,
where 𝜎, 𝜃 are unknown parameters. This thesis is concerned with the estimation
of 𝜎, 𝜃 by using observations
{𝑋(x
1
), 𝑋(x
2
), ⋅⋅⋅ , 𝑋(x
𝑛
)},
where x
1
, ⋅⋅⋅ , x
𝑛
are distinct points in a fixed domain [0, 𝑇]
𝑑
for some constant
0 < 𝑇 < ∞.
Summary vi
Our work has two parts. The first part (Chapter 2) deals with isotropic co-
variance function. Maximum likelihood is a preferred method for estimating the
covariance parameters. However, when the sample size 𝑛 is large, it is a burden
to compute the likelihood. Covariance tapering is an effective technique to ap-
proximating the covariance function with a taper (usually a compactly supported

correlation function) so that the computation can be reduced. Chapter 2 studies
the fixed domain asymptotic behavior of the tapered MLE for the microergodic
parameter of isotropic Mat´ern class covariance function when the taper support is
allowed to shrink as 𝑛 → ∞. In particular when 𝑑 ≤ 3, conditions are established
in which the tapered MLE is strongly consistent and asymptotically normal.
The second part (Chapter 3) establishes consistent estimators of the covariance
and scale parameters of Gaussian random field with multiplicative covariance func-
tion. When 𝑑 = 1, in some cases it is impossible to consistently estimate them
simultaneously under fixed domain asymptotics. However, when 𝑑 > 1, consistent
estimators of functions of covariance and scale parameters can be constructed by
using quadratic variation and spectral analysis. Consequently, they provide the
consistent estimators of the covariance and scale parameters.
vii
List of Tables
Table 2.1 Percentiles (standard errors in parentheses), Means and Stan-
dard Deviations (SD) of
√
𝑛(ˆ𝜎
2
1
𝛼
2𝜈
1,𝑛
−𝜎
2
𝛼
2𝜈
)/(
√
2𝜎

2
𝛼
2𝜈
), and Biases
and Mean Square Errors (MSE) of estimator ˆ𝜎
2
1,𝑛
𝛼
2𝜈
1
for 𝜎 = 1, 𝛼 =
0.8, 𝛼
1
= 1.3, 𝜈 = 1/4. . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Table 2.2 Percentiles (standard errors in parentheses), Means and Stan-
dard Deviations (SD) of
√
𝑛(ˆ𝜎
2
1
𝛼
2𝜈
1,𝑛
−𝜎
2
𝛼
2𝜈
)/(
√
2𝜎

2
𝛼
2𝜈
), and Biases
and Mean Square Errors (MSE) of estimator ˆ𝜎
2
1,𝑛
𝛼
2𝜈
1
for 𝜎 = 1, 𝛼 =
0.8, 𝛼
1
= 1.3, 𝜈 = 1/2. . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Table 2.3 Percentiles (standard errors in parentheses), Means and Stan-
dard Deviations (SD) of
√
𝑛(ˆ𝜎
2
1
𝛼
2𝜈
1,𝑛
−𝜎
2
𝛼
2𝜈
)/(
√
2𝜎

2
𝛼
2𝜈
), and Biases
and Mean Square Errors (MSE) of estimator ˆ𝜎
2
1,𝑛
𝛼
2𝜈
1
for 𝜎 = 1, 𝛼 =
0.8, 𝛼
1
= 2, 𝜈 = 1/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Table 2.4 Percentiles (standard errors in parentheses), Means and Stan-
dard Deviations (SD) of
√
𝑛(ˆ𝜎
2
1
𝛼
2𝜈
1,𝑛
−𝜎
2
𝛼
2𝜈
)/(
√
2𝜎

2
𝛼
2𝜈
), and Biases
and Mean Square Errors (MSE) of estimator ˆ𝜎
2
1,𝑛
𝛼
2𝜈
1
for 𝜎 = 1, 𝛼 =
0.8, 𝛼
1
= 2, 𝜈 = 1/2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
List of Tables viii
Table 2.5 Percentiles (standard errors in parentheses), Means and Stan-
dard Deviations (SD) of
√
𝑛(ˆ𝜎
2
1
𝛼
2𝜈
1,𝑛
−𝜎
2
𝛼
2𝜈
)/(
√

2𝜎
2
𝛼
2𝜈
), and Biases
and Mean Square Errors (MSE) of estimator ˆ𝜎
2
1,𝑛
𝛼
2𝜈
1
for 𝑑 = 1, 𝜎 =
1, 𝛼 = 5, 𝛼
1
= 7.5, 𝜈 = 1/4 and 𝜙
1,1
(𝑥/𝛾
𝑛
). . . . . . . . . . . . . . . 78
Table 2.6 Percentiles (standard errors in parentheses), Means and Stan-
dard Deviations (SD) of
√
𝑛(ˆ𝜎
2
1
𝛼
2𝜈
1,𝑛
−𝜎
2

𝛼
2𝜈
)/(
√
2𝜎
2
𝛼
2𝜈
), and Biases
and Mean Square Errors (MSE) of estimator ˆ𝜎
2
1,𝑛
𝛼
2𝜈
1
for 𝑑 = 1, 𝜎 =
1, 𝛼 = 5, 𝛼
1
= 7.5, 𝜈 = 1/4 and 𝜙
1,1
(𝑥/𝛾
𝑛
). . . . . . . . . . . . . . . 79
Table 2.7 Percentiles (standard errors in parentheses), Means and Stan-
dard Deviations (SD) of
√
𝑛(ˆ𝜎
2
1
𝛼

2𝜈
1,𝑛
−𝜎
2
𝛼
2𝜈
)/(
√
2𝜎
2
𝛼
2𝜈
), and Biases
and Mean Square Errors (MSE) of estimator ˆ𝜎
2
1,𝑛
𝛼
2𝜈
1
for 𝑑 = 2, 𝜎 =
1, 𝛼 = 5, 𝛼
1
= 7.5 and 𝜙
2,1
(x/𝛾
𝑛
). . . . . . . . . . . . . . . . . . . . 80
Table 3.1 means of the estimators and standard errors in parentheses
with 𝜎 = 1, 𝜃
1

= 1, 𝜃
2
= 1.5, 𝛾 = 0.5. . . . . . . . . . . . . . . . . . . 154
Table 3.2 means of
ˆ
𝜎
2
𝜃
1
𝜃
2
and standard errors in parentheses with 𝜎 =
1, 𝜃
1
= 1, 𝜃
2
= 1.5, 𝛾 = 0.5, 𝜆 = 2/3, 𝑛 = 100. . . . . . . . . . . . . . 155
Table 3.3 means of
ˆ
𝜎
2
𝜃
1
𝜃
2
and standard errors in parentheses with 𝜎 =
1, 𝜃
1
= 1, 𝜃
2

= 1.5, 𝛾 = 0.5, 𝜆 = 2/3, 𝑛 = 200. . . . . . . . . . . . . . 155
Table 3.4 means of
ˆ
𝜎
2
𝜃
1
𝜃
2
and standard errors in parentheses with 𝜎 =
1, 𝜃
1
= 1, 𝜃
2
= 1.5, 𝛾 = 1.2, 𝜆 = 2/3, 𝑛 = 200. . . . . . . . . . . . . . 156
ix
List of Figures
Figure 2.1 Histograms of
√
𝑛(ˆ𝜎
2
1,𝑛
𝛼
2𝜈
1
−𝜎
2
𝛼
2𝜈
) without taper for different

𝑑 and 𝑛 with 𝜎 = 1, 𝛼 = 0.8, 𝛼
1
= 1.3, 𝜈 = 1/4. . . . . . . . . . . . . 81
Figure 2.2 Histograms of
√
𝑛(ˆ𝜎
2
1,𝑛
𝛼
2𝜈
1
−𝜎
2
𝛼
2𝜈
) without taper for different
𝑑 and 𝑛 with 𝜎 = 1, 𝛼 = 0.8, 𝛼
1
= 1.3, 𝜈 = 1/2. . . . . . . . . . . . . 82
Figure 2.3 Histograms of
√
𝑛(ˆ𝜎
2
1,𝑛
𝛼
2𝜈
1
−𝜎
2
𝛼

2𝜈
) without taper for different
𝑑 and 𝑛 with 𝜎 = 1, 𝛼 = 0.8, 𝛼
1
= 2, 𝜈 = 1/4. . . . . . . . . . . . . . 83
Figure 2.4 Histograms of
√
𝑛(ˆ𝜎
2
1,𝑛
𝛼
2𝜈
1
−𝜎
2
𝛼
2𝜈
) without taper for different
𝑑 and 𝑛 with 𝜎 = 1, 𝛼 = 0.8, 𝛼
1
= 2, 𝜈 = 1/2. . . . . . . . . . . . . . 84
Figure 2.5 Histograms of
√
𝑛(ˆ𝜎
2
1
𝛼
2𝜈
1,𝑛
−𝜎

2
𝛼
2𝜈
) for different 𝑛 and tapering
range 𝛾
𝑛
= 𝐶𝑛
−0.03
with 𝑑 = 1, 𝜎 = 1, 𝛼 = 5, 𝛼
1
= 7.5, 𝜈 = 1/4. . . . 85
Figure 2.6 Histograms of
√
𝑛(ˆ𝜎
2
1
𝛼
2𝜈
1,𝑛
−𝜎
2
𝛼
2𝜈
) for different 𝑛 and tapering
range 𝛾
𝑛
= 𝐶𝑛
−0.03
with 𝑑 = 1, 𝜎 = 1, 𝛼 = 5, 𝛼
1

= 7.5, 𝜈 = 1/2. . . . 86
Figure 2.7 Histograms of
√
𝑛(ˆ𝜎
2
1
𝛼
2𝜈
1,𝑛
−𝜎
2
𝛼
2𝜈
) for different 𝑛 and tapering
range 𝛾
𝑛
= 𝐶𝑛
−0.02
with 𝑑 = 2, 𝜎 = 1, 𝛼 = 5, 𝛼
1
= 7.5, 𝜈 = 1/4. . . . 87
Figure 2.8 Histograms of
√
𝑛(ˆ𝜎
2
1
𝛼
2𝜈
1,𝑛
−𝜎

2
𝛼
2𝜈
) for different 𝑛 and tapering
range 𝛾
𝑛
= 𝐶𝑛
−0.02
with 𝑑 = 2, 𝜎 = 1, 𝛼 = 5, 𝛼
1
= 7.5, 𝜈 = 1/8. . . . 88
List of Figures x
Figure 2.9 Histograms of
√
𝑛(ˆ𝜎
2
1,𝑛
𝛼
2𝜈
1
− 𝜎
2
𝛼
2𝜈
) for different taper with
𝑑 = 1, 𝜎 = 1, 𝛼 = 0.8, 𝛼
1
= 2, 𝜈 = 1/2. . . . . . . . . . . . . . . . . . 88
xi
LIST Of NOTATIONS

ℝ
𝑑
𝑑-dimensional real space
𝑀
𝑇
the transpose of a matrix 𝑀
∣ ⋅∣ the Euclidean norm i.e. ∣x∣ = (𝑥
2
1
+ ⋅⋅⋅ + 𝑥
2
𝑑
)
1/2
for
x = (𝑥
1
, ⋅⋅⋅ , 𝑥
𝑑
)
𝑇
∈ ℝ
𝑑
∣ ⋅∣
max
∣x∣
max
= max{∣𝑥
1
∣, ⋅⋅⋅ , ∣𝑥

𝑑
∣}, ∀ x = (𝑥
1
, ⋅⋅⋅ , 𝑥
𝑑
)
𝑇
∈
ℝ
𝑑
⌊⋅⌋ ⌊𝑥⌋ denotes the largest integer less than or equal to
𝑥 ∈ ℝ
𝐼{⋅} the indicator function
𝐼
𝑛
𝑛 ×𝑛 identity matrix
List of Notations xii
≲ for real functions 𝑎(𝑥) and 𝑏(𝑥), 𝑎(𝑥) ≲ 𝑏(𝑥) means
that there exists a constant 0 < 𝐶 < ∞ such that
∣𝑎(𝑥)∣ ≤ 𝐶∣𝑏(𝑥)∣ for all possible 𝑥
≍ for real functions 𝑎(𝑥) and 𝑏(𝑥), 𝑎(𝑥) ≍ 𝑏(𝑥) means
that there exist constants 0 < 𝐶
1
< 𝐶
2
< ∞ such
that 𝐶
1
∣𝑏(𝑥)∣ ≤ ∣𝑎(𝑥)∣ ≤ 𝐶
2

∣𝑏(𝑥)∣ for all possible 𝑥
𝑝
−→ converges in probability
a.s.
−→ converges almost sure
𝑑
−→ converges in distribution
1
CHAPTER 1
Introduction
Computer modeling is having a profound effect on scientific research. In de-
terministic computer experiments, unlike physical experiments, no random error
exists, that is, repeating an experiment using the same set of inputs gives the same
response. Suppose 𝑆 ⊂ ℝ
𝑑
is the set of all possible computer inputs, usually called
design space. Let t ∈ 𝑆 denote the 𝑑-dimensional vector of input values and 𝑌 (t)
denote the deterministic response when an exp eriment is run at t. It has become
common practice to approach the deterministic response 𝑌 (t) as a realization of a
spatial process. In this regard, Sacks, Welch, Mitchell and Wynn (1989) modeled
𝑌 (t) as a realization of a Gaussian spatial pro cess (random field), that includes a
2
regression model, called Kriging model,
𝑌 (t) = 𝑋(t) +
𝑝

𝑗=1
𝛽
𝑗
𝑓

𝑗
(t),
where 𝑓
𝑗
(t)’s are known functions, 𝛽
𝑗
’s are unknown coefficients and 𝑋(t) is as-
sumed to be Gaussian random field with mean zero and covariance
𝐶𝑜𝑣(𝑋(t), 𝑋(s)) = 𝜎
2
𝑅(t, s),
where 𝜎
2
is the variance and 𝑅(t, s) is the correlation. The smoothness and prop-
erties of 𝑋(t) are controlled by the correlation function 𝑅(t, s). It is common
to model the data’s variance and correlation structure through a parametric co-
variance function and then estimate these parameters. Thus we will investigate
the asymptotic prop erties of parameter estimators and hope that the asymptotic
results can be applied well as approximations to finite sample.
There are two common asymptotic frameworks in spatial statistics, increas-
ing domain asymptotics and fixed domain asymptotics or called infill asymptotics
(Cressie, 1993). In increasing domain asymptotics, the distance between neighbor-
ing observations is bounded away from zero so that the observation region grows
as the number of observation increases. In fixed domain asymptotics, the num-
ber of observations increases in a given fixed and bounded domain so that the
1.1 Mat´ern Class 3
observations will be increasingly dense. Not surprisingly, the asymptotic behavior
of covariance parameter estimators can b e quite different under these two frame-
works. For example, some covariance parameters are not consistently estimated
under fixed domain asymptotics (Ying, 1991; Zhang, 2004), whereas these same

parameters can be consistently estimated and their maximum likelihood estimators
are asymptotically normal, subject to some regularity conditions, under increasing
domain asymptotics (Mardia and Marshall, 1984).
We are interested in processes on a given fixed region of ℝ
𝑑
and focus on fixed
domain asymptotics. In the next two sections, we will describe some asymptotic
properties under fixed domain asymptotics for two classes of Gaussian processes
with the following covariance functions: Mat´ern class and powered exponential
class respectively.
1.1 Mat´ern Class
There are two kinds of Mat´ern class covariance function. We first introduce
Gaussian process 𝑋(t), t ∈ 𝑆 = [0, 1]
𝑑
with mean zero and isotropic Mat´ern class
covariance function, which was first proposed by Mat´ern in 1960. This covariance
1.1 Mat´ern Class 4
function is defined by
𝐶𝑜𝑣(𝑋(x), 𝑋(y)) =
𝜎
2
(𝛼∣x − y∣)
𝜈
Γ(𝜈)2
𝜈−1
K
𝜈
(𝛼∣x − y∣), (1.1)
∀ x, y ∈ ℝ
𝑑

, where 𝜈 is a known strictly positive constant and 𝜎, 𝛼 are unknown
positive parameters and K
𝜈
is the modified Bessel function of order 𝜈 [see Andrews
et al. (1999), p.223]. The larger 𝜈 is, the smoother 𝑋 is. In particular, 𝑋 will be
𝑚 times mean square differentiable if and only if 𝜈 > 𝑚.
Results for parameter estimation under fixed domain asymptotics are difficult
to derive in general and little work has been done in this area. Stein (1999) strongly
recommended using Gaussian random field with isotropic Mat´ern class covariance
function as a sensible class of models. He investigated the performance of maximum
likelihood estimators for the parameters of a periodic version of the Mat´ern model
with the hope that the large sample results for this periodic mo del would be similar
to those for non-periodic Mat´ern-type Gaussian random fields under fixed domain
asymptotics.
Zhang(2004) proved some important results about isotropic Mat´ern class. He
first pointed out that the induced two mean zero Gaussian measures with (𝜎
2
1
, 𝛼
1
)
and (𝜎
2
0
, 𝛼
0
) are equivalent if 𝜎
2
1
𝛼

2𝜈
1
= 𝜎
2
0
𝛼
2𝜈
0
for 𝑑 ≤ 3. This result consequently
implies that consistent estimators of both 𝜎
2
and 𝛼 cannot exist under fixed domain
1.1 Mat´ern Class 5
asymptotics for 𝑑 ≤ 3. But 𝜎
2
𝛼
2𝜈
can be consistently estimated. In particular,
under some conditions the estimator of 𝜎
2
𝛼
2𝜈
is asymptotically efficient (Zhang,
2004). Furthermore, Kaufman et al. (2008) showed that for any fixed 𝛼, the
estimator of 𝜎
2
𝛼
2𝜈
obtained by tapered maximum likelihood estimator is strongly
consistent under fixed domain asymptotics for 𝑑 ≤ 3 when the taper support is

fixed. Du et al. (2009) showed that such estimator is also asymptotically efficient
for 𝑑 = 1. However, it is unknown whether such estimators lose any asymptotic
efficiency for 𝑑 > 1.
Another kind of Mat´ern class covariance function, called multiplicative Mat´ern
class covariance function, is defined by
𝐶𝑜𝑣(𝑋(x), 𝑋(y)) = 𝜎
2
𝑑

𝑖=1
(𝜃
𝑖
∣𝑥
𝑖
− 𝑦
𝑖
∣)
𝜈
2
𝜈−1
Γ(𝜈)
K
𝜈
(𝜃
𝑖
∣𝑥
𝑖
− 𝑦
𝑖
∣), (1.2)

∀ x = (𝑥
1
, ⋅⋅⋅ , 𝑥
𝑑
)
𝑇
, y = (𝑦
1
, ⋅⋅⋅ , 𝑦
𝑑
)
𝑇
∈ ℝ
𝑑
, where 𝜈 is a known positive constant
and 𝜎
2
, 𝜃
1
, ⋅⋅⋅ , 𝜃
𝑑
are unknown positive parameters and K
𝜈
is the modified Bessel
function of order 𝜈.
By the definitions, we can see that when 𝑑 = 1, multiplicative Mat´ern class
and isotropic Mat´ern class are the same. Thus for multiplicative Mat´ern class, it
follows from Zhang’s (2004) result that 𝜎
2
and 𝜃 cannot be consistently estimated

respectively when 𝑑 = 1. However, the structure of multiplicative Mat´ern class
1.2 Powered Exponential Class 6
for one dimension may be quite different from that for high dimension. When
𝑑 ≥ 2, Loh (2005) showed that for 𝜈 = 3/2 the maximum likelihood estimator
(MLE) of (𝜎
2
, 𝜃
1
, ⋅⋅⋅ , 𝜃
𝑑
) is consistent using observations on a regular grid under
mild conditions. This result implies that 𝜎
2
and 𝜃
1
, ⋅⋅⋅ , 𝜃
𝑑
can be identified for
multiplicative Mat´ern class when 𝑑 ≥ 2 and 𝜈 = 3/2. That also gives an example
to show that the structure of multiplicative Mat´ern class may be quite different
from that of isotropic Mat´ern class for 𝑑 ≥ 2. Thus we cannot directly apply the
results for isotropic Mat´ern class under fixed domain asymptotics to multiplicative
Mat´ern class when 𝑑 ≥ 2.
1.2 Powered Exponential Class
In this section, we introduce Gaussian random field 𝑋(t), t ∈ [0, 1]
𝑑
with
multiplicative powered exponential class covariance function
𝐶𝑜𝑣(𝑋(t), 𝑋(s)) = 𝜎
2

𝑑

𝑖=1
exp{−𝜃
𝑖
∣𝑡
𝑖
− 𝑠
𝑖
∣
𝛾
}, (1.3)
where t = (𝑡
1
, ⋅⋅⋅ , 𝑡
𝑑
)
𝑇
, s = (𝑠
1
, ⋅⋅⋅ , 𝑠
𝑑
)
𝑇
∈ [0, 1]
𝑑
and 𝛾 ∈ (0, 2], 𝜃
1
, ⋅⋅⋅ , 𝜃
𝑑

and
𝜎
2
are strictly positive parameters. The parameter 𝜎
2
is the variance of the process
and the scale parameters 𝜃
1
, ⋅⋅⋅ , 𝜃
𝑑
control how quickly the correlation decays with
distance. Here 𝛾 indicates smoothness of the process. For example, 𝛾 = 1 implies
1.2 Powered Exponential Class 7
that the process is continuous but nowhere differentiable, while 𝛾 = 2 implies that
the process is infinitely differentiable in mean square sense.
For 𝛾 = 1 and 𝑑 = 1, the process known as the Ornstein-Uhlenbeck process
has Markovian properties that for 𝑡 > 𝑠, 𝑋(𝑡) − 𝑒
−𝜃∣𝑡−𝑠∣
𝑋(𝑠) is independent of
𝑋(𝑢), 𝑢 ≤ 𝑠. Another interesting fact for 𝑋(𝑡) is that if 𝜎
2
𝜃 = ˜𝜎
2
˜
𝜃, the in-
duced Gaussian measures with (𝜎
2
, 𝜃) and (˜𝜎
2
,

˜
𝜃) are equivalent [cf. Ibraginov and
Rozanov (1978)]. Thus (𝜎
2
, 𝜃) and (˜𝜎
2
,
˜
𝜃) are not distinguishable with certainty
from the sample path 𝑋(𝑡), that is, it is impossible to estimate consistently both
𝜎
2
and 𝜃 simultaneously. However, Ying (1991) showed that the product ˆ𝜎
2
ˆ
𝜃 of
the maximum likelihood estimators ˆ𝜎
2
and
ˆ
𝜃, as an estimator of 𝜎
2
𝜃, is strongly
consistent and asymptotically normal,
ˆ𝜎
2
ˆ
𝜃
a.s.
−→ 𝜎

2
𝜃,
√
𝑛(ˆ𝜎
2
ˆ
𝜃 − 𝜎
2
𝜃)
𝑑
−→ 𝑁(0, 2(𝜎
2
𝜃)
2
).
As we mentioned before, for 𝑑 = 1, 𝜎
2
and 𝜃 are not identifiable. However, for
𝑑 ≥ 2, under mild conditions Ying (1993) showed that the MLE (ˆ𝜎
2
,
ˆ
𝜃
1
, ⋅⋅⋅ ,
ˆ
𝜃
𝑑
) of
(𝜎

2
, 𝜃
1
, ⋅⋅⋅ , 𝜃
𝑑
) is strongly consistent, i.e.
(ˆ𝜎
2
,
ˆ
𝜃
1
, ⋅⋅⋅ ,
ˆ
𝜃
𝑑
)
a.s.
−→ (𝜎
2
, 𝜃
1
, ⋅⋅⋅ , 𝜃
𝑑
),
1.2 Powered Exponential Class 8
and,
√
𝑛(ˆ𝜎
2

𝑑

𝑖=1
ˆ
𝜃
𝑖
− 𝜎
2
𝑑

𝑖=1
𝜃
𝑖
)
𝑑
−→ 𝑁(0, 2(𝜎
2
𝑑

𝑖=1
𝜃
𝑖
)
2
).
This implies that for 𝑑 ≥ 2, (1.3) gives quite a different structure.
Properties of MLE in other cases were also investigated by some researchers. For
𝛾 = 1, 𝑑 = 2, van der Vaart (1996) showed that the maximum likelihood estimators
are also asymptotically efficient. For 𝛾 = 2, Loh and Lam (2000) showed that
sieve maximum likelihood estimators of 𝜃

1
, ⋅⋅⋅ , 𝜃
𝑑
are strongly consistent using a
regular sampling grid. Unfortunately, when 𝛾 is not an integer, it is rather difficult
to handle likelihood functions analytically, because the covariance matrix of the
data is 𝑛 × 𝑛 and the computations of maximum likelihood can be formidable
as the sample size 𝑛 increases. Also, Mardia and Marshall’s (1984) asymptotic
consistency of MLE is not applicable when the observations are taken from a
bounded region. As a result, properties of MLE under fixed domain asymptotics
are not well understood.
For covariance parameters, the preferred estimator is MLE. However when the
sample 𝑛 is large, it is a burden to compute the likelihood. The first objective of this
thesis is to use covariance tapering to approximate the covariance function so that
the computation can be reduced. Chapter 2 studies the fixed domain asymptotic
behavior of the tapered MLE for the parameter 𝜎
2
𝛼
2𝜈
of isotropic Mat´ern class
1.2 Powered Exponential Class 9
covariance function when the taper supp ort is allowed to shrink as 𝑛 → ∞. The
conditions in Kaufman et al. (2008) are difficult to check in practice. In Chapter
2, conditions will be established in which the tapered MLE is strongly consistent
and asymptotically normal for 1 ≤ 𝑑 ≤ 3.
The second objective of this thesis is to investigate the asymptotic properties
of Gaussian random field with multiplicative covariance function under fixed do-
main asymptotics. We will focus on whether the variance and scale parameters
in multiplicative covariance function can be consistently estimated. In Chapter 3,
we will use quadratic variation and spectral analysis to construct consistent es-

timators. As we mentioned before, the variance and scale parameters cannot be
consistently estimated respectively for one dimension. But the product of these
two parameters can be consistently estimated. The quadratic variation will give
a consistent estimator of this product for 𝑑 = 1 using the observations on a grid.
For 𝑑 > 1, spectral analysis may provide another consistent estimator for product
of variance and all scale parameters. Then these consistent estimators can provide
the consistent estimators of variance and all scale parameters for 𝑑 > 1.
This thesis mainly concerns the consistent estimators of unknown parameters.
It provides closed forms of the estimators. Compared with MLE, these kinds of
estimator are easier to compute and may retain most of the information. However,
all these asymptotic results base on an essential assumption that the smoothness
1.2 Powered Exponential Class 10
parameter is known for technical issue. Some researchers had studied the estima-
tion of smoothness parameter. Stein (1993) studied the estimation of smoothness
parameter for a class of periodic Gaussian processes in one dimension. Constantine
and Hall (1994) studied the estimation of smo othness parameter in one dimension
under a sort of mixture of increasing domain and fixed domain asymptotics. How-
ever, the estimation of smoothness parameter is not the subject to this study.
11
CHAPTER 2
Isotropic Covariance Function
2.1 Introduction
Let 𝑋 : ℝ
𝑑
→ ℝ be a mean-zero isotropic Gaussian random field with the
Mat´ern covariance function
𝐶𝑜𝑣(𝑋(x), 𝑋(y)) = 𝜎
2
𝐾
𝛼

(x −y)
=
𝜎
2
(𝛼∣x −y∣)
𝜈
Γ(𝜈)2
𝜈−1
K
𝜈
(𝛼∣x − y∣), ∀ x, y ∈ ℝ
𝑑
, (2.1)
2.1 Introduction 12
where 𝜈 > 0 is a known constant, 𝛼, 𝜎 are strictly positive but unknown parameters
and K
𝜈
is the modified Bessel function of order 𝜈 [see Andrews et al.(1999), p.223].
It is well known that 𝑋 will be 𝑚 times mean square differentiable if and only if
𝜈 > 𝑚. We refer the reader to Stein (1999) for a comprehensive of Mat´ern-type
Gaussian random fields.
We are concerned with the estimation of the (microergodic) parameter 𝜎
2
𝛼
2𝜈
using observations

𝑋(x
1
), 𝑋(x

2
) ⋅⋅⋅ , 𝑋(x
𝑛
)

,
where x
1
, ⋅⋅⋅ , x
𝑛
are distinct points in [0, 𝑇 ]
𝑑
for some (absolute) constant 0 <
𝑇 < ∞. Zhang (2004) observed it is this microergodic parameter and not the
individual parameters that matters more to interpolation. We refer the reader to
Stein (1999), page 163, for the mathematical definition of microergodicity. For
simplicity, we write
X
𝑛
= (𝑋(x
1
), 𝑋(x
2
) ⋅⋅⋅ , 𝑋(x
𝑛
))
𝑇
.
The covariance matrix of X
𝑛

can be express as 𝜎
2
𝑅
𝛼
, where 𝑅
𝛼
is a 𝑛×𝑛 correlation
matrix whose (𝑖, 𝑗)
th
is element 𝐾
𝛼
(x
𝑖
− x
𝑗
) not depending on 𝜎. Since X
𝑛
∼
2.1 Introduction 13
𝑁
𝑛
(0, 𝜎
2
𝑅
𝛼
), the log-likelihood 𝑙
𝑛
(𝛼, 𝜎) satisfies
𝑙
𝑛

(𝛼, 𝜎) = −
𝑛
2
log(2𝜋) − 𝑛 log(𝜎) −
1
2
log(∣𝑅
𝛼
∣) −
1
2𝜎
2
X
𝑇
𝑛
𝑅
−1
𝛼
X
𝑛
. (2.2)
It is generally acknowledged [e.g., Stein et al. (2004), Furrer et.al. (2006),
Kaufman et al. (2008) and Du et al. (2009)] that in practice, the data set is
usually very large and is irregularly spaced. Computing the inverse covariance
matrix 𝜎
−2
𝑅
−1
𝛼
, which takes 𝑂(𝑛

3
) operations, is then a difficult problem and may
even be intractable in some instances.
A popular and promising way to alleviate this computational problem is to
replace the original covariance function by a more tractable one. More precisely,
we impose a simple (but mis-specified) covariance function for 𝑋 given by
𝐶𝑜𝑣(𝑋(x), 𝑋(y)) = 𝜎
2
1

𝐾
𝛼
1
,𝑛
(x −y), ∀ x, y ∈ ℝ
𝑑
, (2.3)
where

𝐾
𝛼
1
,𝑛
: ℝ
𝑑
→ ℝ is a known isotropic correlation function, 𝛼
1
> 0 is a
known constant and 𝜎
2

1
= 𝜎
2
𝛼
2𝜈
/𝛼
2𝜈
1
.

𝐾
𝛼
1
,𝑛
is allowed to, possibly, vary with
sample size 𝑛. Under (2.3), let 𝜎
2
1

𝑅
𝛼
1
,𝑛
be the covariance matrix of X
𝑛
and hence